Question

show that the equations x+y+z=6, x+2y+3z=14, x+4y+7z=30 are consistent jntu 2002 solution

Answer 1

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Answer 2

Answer:

Given equations are:

    x + y +z  = 6

x + 2y + 3z = 14

 x + 4y + 7z = 30

Consistence condition of linear equation is determinant of coefficients of equation is zero.

$\det \begin{pmatrix}1&1&1\\ 1&2&3\\ 1&4&7\end{pmatrix}$

$\mathrm{Find\:the\:matrix\:determinant\:according\:to\:formula}:$

$\det \begin{pmatrix}a&b&c\\ d&e&f\\ g&h&i\end{pmatrix}=a\cdot \det \begin{pmatrix}e&f\\ h&i\end{pmatrix}-b\cdot \det \begin{pmatrix}d&f\\ g&i\end{pmatrix}+c\cdot \det \begin{pmatrix}d&e\\ g&h\end{pmatrix}$

$=1\cdot \det \begin{pmatrix}2&3\\ 4&7\end{pmatrix}-1\cdot \det \begin{pmatrix}1&3\\ 1&7\end{pmatrix}+1\cdot \det \begin{pmatrix}1&2\\ 1&4\end{pmatrix}$

$\det \begin{pmatrix}2&3\\ 4&7\end{pmatrix}=2$

$\det \begin{pmatrix}1&3\\ 1&7\end{pmatrix}=4$

$\det \begin{pmatrix}1&2\\ 1&4\end{pmatrix}=2$

$=1\cdot \:2-1\cdot \:4+1\cdot \:2$

$1\cdot \:2-1\cdot \:4+1\cdot \:2=0$

=0

Hence, equations are: x + y +z  = 6 ,  x + 2y + 3z = 14 and  x + 4y + 7z = 30 are consistent